Landau and Bloch theorems for harmonic mappings

Chen, Gauthier and Hengartner obtained some versions of Landau's theorem for bounded harmonic mappings and Bloch's theorem for harmonic mappings which are quasiregular and for those which are open. Later, Dorff and Nowak improved their estimates concerning Landau's theorem. In this study, we improve these last results by obtaining sharp coefficient estimates for properly normalized harmonic mappings. Furthermore, our estimates allow us to improve Bloch constant for open harmonic mappings.     

A Schwarz Lemma for Harmonic Mappings Between the Unit Balls in Real Euclidean Spaces

The authors prove a Schwarz lemma for harmonic mappings between the unit balls in real Euclidean spaces. Roughly speaking, this result says that under a harmonic mapping between the unit balls in real Euclidean spaces, the image of a smaller ball centered at origin can be controlled. This extends the related result proved by Chen in complex plane.     

Harmonic mappings onto stars

A general version of the Radó-Kneser-Choquet theorem implies that a piecewise constant sense-preserving mapping of the unit circle onto the vertices of a convex polygon extends to a univalent harmonic mapping of the unit disk onto the polygonal domain. This paper discusses similarly generated harmonic mappings of the disk onto nonconvex polygonal regions in the shape of regular stars. Calculation of the Blaschke product dilatation allows a determination of the exact range of parameters that produce univalent mappings.     

Harmonic mappings with hereditary starlikeness

We study a hereditary starlikeness property for planar harmonic mappings on a disk and on an annulus. While such a property is a common trait of conformal mappings, it may be absent in harmonic mappings. It turns out that a sufficient condition for a harmonic mapping f to possess this hereditary property is to have a harmonic argument — a striking feature of conformal mappings that does not extend to all harmonic mappings.     

Hölder continuity of quasiconformal harmonic mappings from the unit ball to a spatial domain with C1 boundary

We prove that every quasiconformal mapping from the harmonic $\beta$-Bloch space between the unit ball and a spatial domain with $C^1$ boundary is globally $\alpha$-Hölder continuous for $\alpha <1?\beta$, with the Hölder coefficient that does not depend neither on the mapping nor on $\beta$. An analogous result also holds for Lipschitz continuous, quasiconformal harmonic mappings for $\alpha <1$. This is an approach towards the extension of some results from the complex plane obtained by Warschawski (1951) for conformal mappings and Kalaj (2022) for quasiconformal harmonic mappings.     

On asymptotically nonexpansive mappings in hyperconvex metric spaces

Since bounded hyperconvex metric spaces have the fixed point property for nonexpansive mappings, it is natural to extend such a powerful result to asymptotically nonexpansive mappings. Our main result states that the approximate fixed point property holds in this case. The proof is based on the use, for the first time, of the ultrapower of a metric space.

     

Estimates on Bloch constants for planar harmonic mappings

The Schwarz lemma and Bloch constants for planar bounded harmonic mappings are considered. Sharper form and better estimates are obtained. Our results improve the one made by Dorff and Nowak as well as by Chen, Gauthier and Hengartner.     

On the Douglas–Dirichlet functional and harmonic quasiconformal mappings

In this article, we discuss the minimal mappings of Douglas–Dirichlet functional and harmonic quasiconformal mappings, and solve the uniqueness problem of harmonic quasiconformal mappings posed by Shibata.     

On Quasimonotone Variational Inequalities

In this paper, we study variational inequalities with multivalued mappings. By employing Fan's lemma, we establish the existence result for the dual formulation of variational inequalities with semistrictly quasimonotone mappings. We also show that similar results for quasimonotone variational inequalities do not hold.     

Boundary behaviour of the mappings satisfying generalized inverse modular inequalities

We study the geometric properties of the mappings for which generalized inverse modular inequalities hold. We generalize in this way known theorems from the theory of analytic mappings and the theory of quasiregular mappings, like the theorems of Fatou, M. and F. Riesz, Beurling and Lindelöf and their extensions given for quasiregular mappings by Martio, Rickman and Vuorinen.     

New fixed point theorems of condensing mappings satisfying the interior condition in Banach spaces

In this paper, based on a basic result on condensing mappings satisfying the interior condition, some new fixed point theorems of the condensing mappings of this kind are obtained. As a result, the famous Altman’s theorem, Roth’s theorem and Petryshyn’s theorem are extended to condensing mappings satisfying the interior condition.     

Almost biased mappings and almost compatible mappings are equivalent under some condition

The concepts of almost biased mappings and almost compatible mappings are introduced. Some variants of these concepts are also discussed. A common fixed point theorem for a pair of almost I-biased mappings in metric spaces is proved under certain condition. In the sequel, it is shown that almost biased mappings and almost compatible mappings are equivalent under some condition.     

Duality for Set-Valued Multiobjective Optimization Problems,Part 1: Mathematical Programming

The duality of multiobjective problems is studied with the help of the apparatus of conjugate set-valued mappings introduced by the author. In this paper (Part 1), a duality theory is developed for set-valued mappings, which is then used to derive dual relations for some general multiobjective optimization problems which include convex programming and optimal control problems. Using this result, in the companion paper (Part 2), duality theorems are proved for multiobjective quasilinear and linear optimal control problems. The theory is applied to get dual relations for some multiobjective optimal control problem.     

多复变数双全纯映照的一些偏微分不等式

本文在有界平衡域上建立一些与双全纯映照有关的偏微分不等式．作为应用，给出 有界平衡域上星形映照以及近似星形映照的一些充分判别条件．     

关于局部紧度量空间的映象

本文借助于商映射、伪开映射和闭映射建立局部紧度量空间和几类具有某些特定性质mk-系之间的联系，作为推论.得到：双商ss-映射保持局部紧度量空间.     

A note on harmonic quasiconformal mappings

In this note we show that a harmonic quasiconformal mapping f=u+iv with respect to the Poincaré metric of the upper half plane onto itself such that v(x,y)=v(y) or u(x,y)=u(x) is a conformal mapping.     

Common fixed point theorems for weakly compatible quasi contraction mappings

Summary Recently, Pathak [13] has made an extension of the notion of compatibility to weak compatibility, and extended the coincidence theorem for compatible mappings in Kaneko and Sessa [11] to weakly compatible mappings [13]. In the present paper, we define a new class of weakly compatible mappings (Definition 4) and prove some common fixed point theorems for these mappings, which satisfy Condition (2) below. Although our main theorem is formulated for weakly compatible mappings, its corresponding formulation for commutative mappings is also a new result, thus presenting a generalization of some theorems of Fisher, Das and Naik, Khan and Kubiaczyk, Reich, Ćirić and Rhoades and Watson.     

An optimization result for set-valued mappings and a stability property in vector problems with constraints

First, we prove an existence result relative to minimal points of set-valued mappings. Then, conditions about the upper and lower semicontinuity of constraint sets defined through set-valued mappings are given. Finally, a stability result relative to vector problems with abstract constraints is proved.The author thanks the referee for helpful comments on the first version of this paper.     

A common fixed point theorem for compatible quasi contractive self mappings in metric spaces

A common fixed point theorem for weakly commuting quasi contractive self mappings with contracting orbital diameters in metric spaces [V. Berinde, A common fixed point theorem for quasi contractive type mappings, Ann. Univ. Sci. Budapest. 46 (2003) 81-90] is extended to the more general class of compatible quasi contractive self mappings. Our result does extend and generalize numerous related results in literature.     

有界凸圆型域上的准凸映照

本文研究星形映照的一类子族——有界凸圆型城上的准凸映照,并讨论了它与星形映照及凸映照的关系,建立其与凸映照同形的增长定理和掩盖定理．